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Geometric Sequences and Series
A sequence is geometric if the ratios of consecutive terms are the same.
2, 8, 32, 128, 512, . . .
geometric sequence
The common ratio, r, is 4.
82
4
328
4
12832
4
512128
4
Example 1.
a. Is the sequence geometric? If so, what is ?
2,4,8,16,...2 ,...n
r
4 8 162, 2, 2
2 4 8 2r
b. Is the sequence geometric? If so, what is ?
1 1 1 1 1 , , , ,..., ,...
3 9 27 81 3
n
r
1 1 11 1 19 27 81
1 1 13 3 33 9 27
1
3r
The nth term of a geometric sequence has the form
an = a1rn - 1
where r is the common ratio of consecutive terms of the sequence.
The nth Term of a Geometric Sequence
15, 75, 375, 1875, . . . a1 = 15
The nth term is 15(5n-1).
75 515
r
a2 = 15(5)
a3 = 15(52)
a4 = 15(53)
1Example 2. Write the first five terms of the geometric sequence whose first term is a 3
and whose common ratio is 2.r
1
12
3
3 2 6
a
a
2
3 3 2 12a 3
4 3 2 24a 4
5 3 2 48a
Example 3. Find the 15th term of the geometric sequence whose first term is 20 and whose common ration is 1.05.
11n
na a r
14
15 20 1.05a 39.599
Example 4. Find a formula for the nth term of the following geometric sequence. What is the ninth term of the sequence?
5, 15, 45, …
Find the common ratio
15 5 3 45 15 3
15 3
n
na
8
9 5 3a 32,805
125Example 5. The 4th term of a geometric sequence is 125, and the 10th term is .
64Find the 14th term. Assume that the terms of the sequence are positive.
10 410 4a a r
6125125
64r
61
64r
1
2r
414 10a a r
4125 1
64 2
125
1024
The sum of the first n terms of a sequence is represented by summation notation.
Definition of Summation Notation
1 2 3 41
n
i ni
a a a a a a
index of summation
upper limit of summation
lower limit of summation
5
1
4n
n
1 2 3 4 54 4 4 4 4 4 16 64 256 1024 1364
The Sum of a Finite Geometric Sequence
The sum of a finite geometric sequence is given by
11 1
1
1 .1
n nin
i
rS a r ar
5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ?
n = 8
a1 = 5
1
81 11
221
5n
nrS ar
5210r
1 25651 2 2555
1 1275
12
1
Example 6. Find the sum 4 0.3n
n
Write out a few terms.
12
1 2 3 12
1
4 0.3 4 0.3 4 0.3 4 0.3 ... 4 0.3n
n
1 4 0.3 0.3 and 12a r n
12
11
14 0.3
1
nn
n
ra
r
121 0.3
4 0.31 0.3
1.714
If the index began at i = 0, you would have to adjust your formula
12 12
0
0 1
4 0.3 4 0.3 4 0.3n
n
i n
12
1
4 4 0.3n
n
4 1.714 5.714
Definition of Geometric Series
The sum of the terms of an infinite geometric sequence is called a geometric series.
a1 + a1r + a1r2 + a1r3 + . . . + a1rn-1 + . . .
If |r| < 1, then the infinite geometric series
11
0
.1
i
i
aS a r
r
has the sum
If 1 , then the series does not have a sum.r
Example 7. Use a graphing calculator to find the first six partial sums of the series. Then find the sum of the series.
1cum sum(seq(4*0.6 , ,1,6))x x
4, 6.4, 7.84, 8.704, 9.2224, 9.53344
Use the formula for the sum of an infinite series to find the sum.
410
1 0.6S
1
1
4 0.6n
n
Example 8. Find the sum of 3 + 0.3 + 0.03 + 0.003 + …,
33.33
1 0.1S
Example 9. A deposit of $50 is made on the first day of each month in a savings account that pays 6% compounded monthly. What is the balance at the end of 2 years?
This type of savings plan is called an increasing annuity.
The first deposit will gain interest for 24 months, and its balance will be
The second deposit will gain interest for 23 months
24
24
24
0.0650 1 50 1.005
12A
23
23
23
0.0650 1 50 1.005
12A
The last deposit will gain interest for only one month
1
1
0.0650 1 50 1.005
12A
The total balance will be the sum of the balances of the 24 deposits.
24
1
1 1.005150 1.005 $1277.96
1 1 1.005
n
n
rS a
r
Homework
Page 607-608
2-24 even, 25-31 odd, 33, 35, 40, 42, 48-54 even